Contractivity of time-dependent driven-dissipative systems

In a number of physically relevant contexts, a quantum system interacting with a decohering environment is simultaneously subjected to time-dependent controls and its dynamics is thus described by a time-dependent Lindblad master equation. Of particular interest in such systems is to understand the circumstances in which, despite the ability to apply time-dependent controls, they lose information about their initial state exponentially with time i.e., their dynamics are exponentially contractive. While there exists an extensive framework to study contractivity for time-independent Lindbladians, their time-dependent counterparts are far less well understood. In this paper, we study the contractivity of Lindbladians, which have a fixed dissipator (describing the interaction with an environment), but with a time-dependent driving Hamiltonian. We establish exponential contractivity in the limit of sufficiently small or sufficiently slow drives together with explicit examples showing that, even when the fixed dissipator is exponentially contractive by itself, a sufficiently large or a sufficiently fast Hamiltonian can result in non-contractive dynamics. Furthermore, we provide a number of sufficient conditions on the fixed dissipator that imply exponential contractivity independently of the Hamiltonian. These sufficient conditions allow us to completely characterize Hamiltonian-independent contractivity for unital dissipators and for two-level systems.